Section 9.2 of Art of Problem Solving’s Introduction to Algebra is one of my favorite sections in any book that my kids have gone through. The section has the simple title – “Which is Greater?”

One question from that section that was giving my younger son some trouble today was this one:

Which is greater or

I decided our conversation about the problem would make a great Family Math talk, so we dove in – his first few strategies to try to solve the problem resulted in dead ends, unfortunately. By the end of the video, though, we had a strategy.

Now that we’d found that and are close together, we tried to use that idea to find out more information about the original numbers.

I found his idea of approximating at the end to be fascinating even if it wasn’t quite right. It was also interesting to me how difficult it was for him to see that the two numbers on the left hand side of the white board were each bigger than the two corresponding numbers on the right hand side of the board. It is such a natural argument for someone experienced in math, but, as always, it is nice to be reminded that arguments like that are not obvious to kids.

I’m wrapping up sequences and series this week with my son and the final topic is Taylor Series. We’d had a few discussions here and there about power series, but it all comes together this week. Looking through some old problems form the BC calculus test, I found a nice one from 2011 that I wanted to use to introduce the idea of error terms.

I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for and then write down the series for

The next question asks you to write the series for and then write the Taylor Series around x = 0 for the function .

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week:

My younger son is beginning to study quadratic equations in Art of Problem Solving’s Introduction to Algebra book. So far he’s essentially only seen quadratic equations that factor over the integers. For today’s project I wanted to show him that there are simple equations with fairly complicated (compared to integers!) roots.

We started with a problem similar to ones that he’s already seen:

Next I showed him a type of equations that he’s not see before and we spent 5 min talking about his ideas of how you could solve it:

Finally, for the specific equation we were looking at, I showed him how we could use continued fractions to solve it. As a bonus he remembered the connection between the Fibonacci numbers and the golden ratio and that got us to the exact solution!

Today I shared the program with the boys. It has 4 different types of random walks to explore. For each one I asked the boys what they thought would happen. At the end we looked at all 4 simultaneously.

Sorry that the starting videos are so blue – I didn’t notice that while we were filming (and didn’t do anything to fix it, so I don’t know why the last two vides are better . . . .)

Also, following publication, I learned the author of the program we were playing with:

Last week Grant Sanderson published an incredible video about the surface area of a sphere:

By happy coincidence my older son is spending a little time reviewing polar coordinates now. Although not exactly the same ideas, I think there’s enough overlap to make studying Sanderson’s new video worthwhile.

So, I’m going to do a 6-part project going through the video. Tonight we watched it and my son’s initial thoughts are below. Each of the next 5 parts will be spent discussing and answering the 5 questions that Sanderson asks in the 2nd half of the video.

Here’s question #1:

We’ve been away from right triangle trig a little bit lately, so I was interested to see how my son would approach this problem. His approach was a bit of a surprise, but it did get him to the right answer:

The next question in Grant’s video is about how the area of one of the rings on the sphere changes when you project it down to the “base” of the sphere (see the picture above).

I thought that answering this question would be a really good geometry, trig, and Calculus exercise for my son:

Now we get to a really interesting part -> Question #3

Grant asks you to relate the area you found in question 2 – the area of a ring around the sphere projected down to the center of the sphere – to the area of a different ring around the sphere.

Here’s my son’s work on this problem:

Finally – my son answers questions #4 and #5 after a quick review of the previous results. He was a little tired tonight, but we needed to squeeze in these two questions tonight because I have to travel for work tomorrow.

Tactile is a modern reimplementation of my PhD code for manipulating isohedral tilings, now available as open-source libraries in C++ and Javascript. I've also created a few fun web-based apps for playing with tilings. https://t.co/vnt8hLIPbl

Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day 🙂

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!

We are about to start the section on power series, but since I haven’t blogged about our calculus work in a while I wanted to do a blog post about testing convergence of series. I chose this problem from the 2012 BC calculus exam:

Here is my son’s work on the first series:

Here’s his work on the 2nd series:

Here’s his work on the 3rd series:

This has been a fun topic to cover. I’m excited to start on power series tomorrow!